Virgil
Posts:
8,833
Registered:
1/6/11


Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 4:31 PM


In article <kc6idc$jc$1@Kilnws1.UCIS.Dal.Ca>, gus gassmann <gus@nospam.com> wrote:
> On 03/01/2013 5:53 PM, Virgil wrote: > > In article > > <a60601d524a24501a28b84a7b1e53bac@ci3g2000vbb.googlegroups.com>, > > WM <mueckenh@rz.fhaugsburg.de> wrote: > > > >> On 3 Jan., 14:52, gus gassmann <g...@nospam.com> wrote: > >> > >>> Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT* > >>> reals r1 and r2, then you can establish this fact in finite time. > >>> However, if you are given two different descriptions of the *SAME* real, > >>> you will have problems. How do you find out that NOT exist n... in > >>> finite time? > >> > >> Does that in any respect increase the number of real numbers? And if > >> not, why do you mention it here? > > > > It shows that WM considerably oversimplifies the issue of > > distinguishing between different reals, or even different names for the > > same reals. > >>> > >>> Moreover, being able to distinguish two reals at a time has nothing at > >>> all to do with the question of how many there are, or how to distinguish > >>> more than two. Your (2) uses a _different_ concept of distinguishability. > >> > >> Being able to distinguish a real from all other reals is crucial for > >> Cantor's argument. "Suppose you have a list of all real numbers ..." > >> How could you falsify this statement if not by creating a real number > >> that differs observably and provably from all entries of this list? > > > > Actually, all that is needed in the diagonal argument is the ability > > distinguish one real from another real, one pair of reals at a time. > > Exactly. The only reals that matter to Cantor's argument are the > *countably* many that are assumed to have been written down. There is no > need (nor indeed an effective way) to distinguish the constructed > diagonal from *all* the potential numbers that could have been > constructed that are not on the list, either. Any *one* number not on > the list shows that the list is incomplete and thus establishes the > uncountability of the reals.
But try getting WM to see it! 

